Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $f$ is a root of $g,$ and the $x$-coordinate of the vertex of $g$ is a root of $f,$ and both $f$ and $g$ have the same minimum value.  If the graphs of the two polynomials intersect at the point $(100,-100),$ what is the value of $a + c$?
Explanation: By symmetry, the line $x = 100$ must be equidistant to both vertices of the parabolas.  Furthermore, the $x$-coordinate of the vertex of $f$ is $-\frac{a}{2},$ and the $x$-coordinate of the vertex of $g$ is $-\frac{c}{2}.$

[asy]
unitsize(2 cm);

real parabone (real x) {
  return (x^2 - 1);
}

real parabtwo (real x) {
  return ((x - 1)^2 - 1);
}

draw((-1.2,0)--(2.2,0));
draw(graph(parabone,-1.2,1.2),red);
draw(graph(parabtwo,-0.2,2.2),blue);
draw((0,0)--(0,-1),dashed);
draw((1,0)--(1,-1),dashed);

label("$y = f(x)$", (-1.2,parabone(1.2)), N, red);
label("$y = g(x)$", (2.2,parabtwo(2.2)), N, blue);

dot((0,0));
dot((0,-1));
dot((1,0));
dot((1,-1));
[/asy]

Therefore,
\[\frac{-\frac{a}{2} - \frac{c}{2}}{2} = 100,\]which implies $a + c = \boxed{-400}.$